Before we state the version of the Atiyah-Singer index formula we will need, let us recall some basic facts on classifying spaces. Let \(G\)  be a compact Lie group and let \(EG\rightarrow BG\)  denote the universal principal \(G\) -bundle over the classifying space \(BG\) . If \(\mathcal {P} \rightarrow M\)  is a principal \(G\) -bundle, then there is up to homotopy a unique map \(f\colon M \rightarrow BG\)  such that \(\mathcal {P}\cong f^{*}EG\) , the pullback bundle. In cohomology, \(f\)  induces a ring homomorphism \(f^{**}\colon H^{**}(BG;\mathbb {Q}) \rightarrow H^{**}(M;\mathbb {Q})\) , where \(H^{**}(-;\mathbb {Q}) = \prod _{k\ge 0}H^{k}(-;\mathbb {Q})\) . Finally, it is shown in [1]} that \(H^{**}(BG;\mathbb {Q})\)  is a ring of formal power series in several indeterminates with rational coefficients, hence an integral domain. In the following, \(\operatorname{\mathrm {ch}}\)  denotes the Chern character and \(\operatorname{\mathrm {td}}\)  the Todd class.
